Asymptotic normality and optimality in nonsmooth stochastic approximation

Abstract

In their seminal work, Polyak and Juditsky showed that stochastic approximation algorithms for solving smooth equations enjoy a central limit theorem. Moreover, it has since been argued that the asymptotic covariance of the method is best possible among any estimation procedure in a local minimax sense of Hájek and Le Cam. A long-standing open question in this line of work is whether similar guarantees hold for important non-smooth problems, such as stochastic nonlinear programming or stochastic variational inequalities. In this work, we show that this is indeed the case.

The arxiv report arXiv:2108.11832 has been split into two parts. This is Part 2 of the original submission, augmented by a some new results and a reworked exposition